3.1 linear equations: one transformation linear equations in one variable a linear equation in one...
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![Page 1: 3.1 Linear Equations: One Transformation Linear Equations in One Variable A linear equation in one variable is an equation that can be written in the form](https://reader036.vdocuments.mx/reader036/viewer/2022070414/5697c0121a28abf838ccc4e7/html5/thumbnails/1.jpg)
3.1 Linear Equations: One TransformationLinear Equations in One Variable
A linear equation in one variable is an equation that can be written in the form
ax + by = c, where a, b, and c are real numbers and a ≠ 0.An equation is a statement with an equals sign.
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3.1 Linear Equations:One Transformation
Original
Equation
Equivalent
Equation
1. Add the same number to both sides. x - 4 = 7 x = 11
2. Subtract the same number to both sides. x + 2 = 9 x = 7
3. Multiply both sides by the same number. x/2 = 4 x = 8
4. Divide both sides by the same number. 5x = 20 x = 4
5. Interchange the two sides. x = 7 7 = x
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3.1 Linear Equations:One Transformation Two equations are equivalent if they have the
same solution set. In solving an equation, the goal is to isolate the
variable by using inverse operations.
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3.1 Linear Equations:One Transformation
Solve x + 3 = 7 x + 3 - 3 = 7 - 3 Add -3 to both sides x = 4 Simplify
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3.1 Linear Equations:One Transformation
Solve 3x = -12
3x
3
12
3 Divide both sides by 3
x 4 Simplify
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3.1 Linear Equations:One Transformation
Solve -4 = x - 7
4 + 7 = x - 7 + 7 Add 7 to both sides.
3 x Simplify
x 3 Transpose
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3.1 Linear Equations:One TransformationSolve
x
5= -9
x
5
5 = -9(5) Multiply both sides by 5.
x 45 Simplify
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3.1 Linear Equations:One Transformation
Properties of Equalities
1. If a = b, then a + c, = b + c Addition Property
2. If a = b, then a - c, = b - c Subtraction Property
3. If a = b, then ac = bc Multiplication Property
4. If a b and c 0, then a
c
b
c Division Property
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3.2 Linear Equations: Two or More Transformations Using Two or More Transformations
1. Simplify both sides of the equation.
2. Use inverse operations to isolate the variable.
3. Check the solution.
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3.2 Linear Equations:Two or More TransformationsSolve
4x + 7 - 2x =13
2x 7 13 Combine like terms.
2x 6 Subtract 7 from both sides.
x 3 Divide both sides by 2.
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3.2 Linear Equations:Two or More TransformationsSolve
5x +2(3 - x) =15
5x 6 2x 15 Remove parentheses
3x 6 15 Combine like terms.
3x 9 Subtract 6 from both sides.
x 3 Divide both sides by 3.
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3.2 Linear Equations: Two or More Transformations
Solve: 3x + 6 = 12
3x + 6 - 6 = 12 – 6 Add -6 to both sides
3x = 6 Combine like terms
x = 2 Divide both sides by 3
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3.2 Linear Equations: Two or More Transformations
Solve: 4.5 = 3 + 2x
4.5 - 3 = 3 - 3 + 2x Add -3 to both sides
1.5 = 2x Divide both sides by 2
.75 = x
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3.2 Linear Equations: Two or More Transformations
Solve: -2x + 2 - 4x = 20
-6x + 2 = 20 Combine like terms
-6x + 2 – 2 = 20 – 2 Add -2 to both sides
-6x = 18 Divide by -6
x = -3
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3.2 Linear Equations: Two or More Transformations
Solve: 4(x - 2) = -10
4x - 8 = -10 Remove parenthesis
4x – 8 + 8 = -10 + 8 Add 8 to both sides
4x = -2 Combine like terms
x = -2/4 = -1/2 Divide both sides by -4
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3.2 Linear Equations: Two or More Transformations
Solve:
-3(x - 2) = 21
-3x + 6 = 21 Remove parenthesis
-3x + 6 - 6 = -21 - 6 Add -6 to both sides
-3x = -27 Combine like terms
x = 9 Divide both sides by -3
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3.3 Solving Equations with Variables on Both Sides
Collect variables on the side with the greatest variable coefficient
3x 5 8x 30
3x 5 3x 8x 30 3x
5 5x 30
35 5x
7 x
3x 5 8x 30
3x 5 8x 8x 30 8x
5x 5 30
5x 35
x 7
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3.3 Solving Equations with Variables on Both Sides
Solve:
4(y 2) 6y 2 8y
4y 8 2y 2
4y 8 2y 2y 2 2y
6y 8 2
6y 8 8 2 8
6y 6
y 1
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3.3 Solving Equations with Variables on Both Sides
Solve:
3(x 5) 2x 10 4x
3x 15 2x 10
5x 15 10
5x 5
x 1
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3.4 Problem SolvingGeneral Strategies for Problem Solving
1. Read and understand the problem.Choose a variableConstruct a diagram
2. Translate the problem into an equation. 3. Solve the equation. 4. Interpret the results.(Check your answer)
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3.5 Solving Equations That Involve Decimals
12.3x - 5.1 = 17
Solve:
12.3x - 5.1+5.1 = 17 + 5.1
12.3x = 22.1
x 22.1
12.3
x 1.796747
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3.6 Literal Equations
Solve for v d =vt
Solve for l 2w +2l =P
2l =P - 2w
Solve for x y = mx + b
l P 2w
2
y b mx
y b
mx
d
tv
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3.7 Scatter Plots
Quadrant IQuadrant II
Quadrant III Quadrant IV
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3.7 Scatter Plots
A(3,2)
BB(-1,-3)A
C(-5,0)B